I would like to know that for what real value of $p$, the infinite product $$\prod_{n=1}^\infty \left(1 + \frac{i (-1)^n}{n^p} \right)$$ converges.
I only know that $\prod z_n$ converges if and only if $\sum \log z_n$ converges if $\Re z_n > 0$. It looks like the alternating series for me, but I'm not sure how to proceed.
Also, can the convergence series tests in the real case (e.g, comparison test, integral test,...) be applied to complex series?
Thank you.